A classical theorem of Koksma states that for Lebesgue almost every x > 1 the sequence (xⁿ) ^∞_n=1 is uniformly distributed modulo one. In the present paper we extend Koksma’s theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every x > 1 the pair correlations of the fractional parts of (xⁿ) ^∞_n=1 are asymptotically Poissonian. The proof is based on a martingale approximation method.

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关于实数幂的对相关

Koksma的一个经典的定理指出，对于几乎勒贝格每X > 1的序列（X ^Ñ）^∞ _{ñ = 1}是均匀分布的模之一。在本文中，我们将Koksma定理扩展到对相关设置。更确切地说，我们表明，对于几乎勒贝格每X > 1个的一对（的小数部分的相关性X ^Ñ）^∞ _{ñ = 1}渐近泊松。该证明基于a近似方法。